Dear categorists (and topologists), it is known (Clementino, Hofmann, Tholen, Richter, Niefield...) that perfect maps p:X->Y are exponentiable in Top/Y, and that the same holds for local homeomorphisms h:X->Y. Question: is it true that 1) p=>h is a local homeomorphism 2) h=>p is a perfect map? The conjecture is suggested by the following observations: A) it holds both for Y = 1 (compact and discrete space) and for subspaces inclusion (closed and open parts) B) the analogy between perfect maps and local homeomorphisms with discrete (op)fibrations (via convergence or other considerations) and the fact that 1) and 2) above hold in Cat/Y: if p is a discrete fibration and h is a discrete opfibration then p=>h is itself a discrete opfibration (and conversely). Best regards, Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]